$$T_U = \hbar a/(2\pi c k_B) \sim 3\times10^3\,\mathrm K$$

Equation 105 · Evolutionary Physics

No Particle Without a Cosigner

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Electrons circulating in a storage ring undergo enormous proper acceleration — for an ultra-relativistic beam of Lorentz factor $\gamma$ on a ring of radius r , the proper centripetal acceleration is a $\approx$ $\gamma^2$ $c^2$/r . For beam parameters typical of a large electron-positron collider, with $\gamma$ of order 2$\times$10^5 and r of order several kilometers, this works out to a $\sim$ 8$\times$10^{23}\,$\mathrm{m\,s^{-2}}$ , corresponding by the Unruh formula to $T_U$ = $\hbar$ a/(2$\pi$ c $k_B$) $\sim$ 3$\times$10^3\,$\mathrm K$ — a temperature scale that is, remarkably, not astronomically small, unlike the roughly 10^{-19}\,$\mathrm K$ an Earth-bound laboratory accelerometer would need a planet’s worth of…

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